Families of Ramanujan Graphs and Quaternion Algebras

Expander graphs are graphs in which the neighbors of any given “not too large” set of vertices X form a large set relative to the size of X—rumors tend to spread very fast. Among those, the Ramanujan graphs are extremal in their expansion properties. To be precise, the eigenvalues of the adjacency matrix have an extremal property that guarantees good expansion properties. Expanders, and hence Ramanujan graphs, have many applications, practical and theoretical, to computer science, coding theory, cryptography and network construction, besides numerous purely mathematical applications. Some applications are briefly indicated in the last section of this paper; for a thorough overview see [14, 38] and the references therein. Quaternion algebras make an appearance in many constructions of Ramanujan graphs. The constructions of Lubotzky – Phillips – Sarnak [24, 25] and Pizer [31] have used definite quaternion algebras over Q, Pizer’s construction allowing a more general setting, while making the arithmetic of quaternion algebras more dominant. The construction by Jordan –Livné [18, 23] makes use of quaternion algebras over totally real fields, but in essence is built out of the LPS (for Lubotzky, Phillips, Sarnak) graphs. In each of these cases the Ramanujan property follows from the Ramanujan conjecture for a suitable space of automorphic representations. This much is true also for a related construction by Li [21]. In hindsight, given recent research into Ramanujan complexes (see, e.g., [4,26]), the reason for the appearance of quaternion algebras is that they supply one with discrete co-compact subgroups of PGL2(F ), where F is a finite extension of Qp. The combinatorial properties of the graphs, or, more generally, complexes, constructed from the Bruhat – Tits buildings associated PGLn(F ), are intimately related to automorphic forms on the appropriate group. The construction presented in this paper generalizes some of Pizer’s work from definite quaternion algebras over Q to totally definite quaternion algebras over totally real fields. It is, in essence, a special case of the construction by Jordan – Livné (JL), though our main examples are different from theirs as our emphasis